[FOM] 583: Continuation Theory 2

Tue Jun 23 12:01:01 EDT 2015

The paper https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87   has title    Mathematically Natural Concrete Incompleteness, and
will be retitled

Mathematically Natural Concrete Incompleteness - order invariant sets.

The Continuation Theory is expected to have its own special
motivations and directions, and so I am expecting a second paper with
the title

Mathematically Natural Concrete Incompleteness - continuations.

Recall the following statements, the first from Order Invariant Sets,
the second from Continuations. They are very closely related, almost
the same, but not quite.

EVERY ORDER INVARIANT R CONTAINEDIN Q^2k HAS A MAXIMAL NONNEGATIVE
ROOT, WHERE S_1...n|>n = S_0...n-1|>n.

EVERY FINITE SUBSET OF Q^k|>n HAS A MAXIMAL NONNEGATIVE CONTINUATION,
WHERE S_1...n|>n = S_0...n-1|>n.

Both are provably equivalent to Con(SRP) over WKL_0, and hence
independent of ZFC.

All definitions are clearly in

[1] https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87.

Also the definitions for the second are given in

[2] http://www.cs.nyu.edu/pipermail/fom/2015-June/018789.html .

Since we are in the realm of perfection, we will try to be very
demanding - more or even much more so than for a lot of good solid
contemporary mathematics.

Here are some advantages for the second formulation:

1. Only k dimensional sets are used. No detour into a 2k dimensional set.
2. The conceptual motivation for roots and maximizing them in the
first statement can be questioned. In the second statement, roots are
not formed directly, as they are the continuations. Maximizing
continuations is very natural mathematically: go as far as possible
given a certain constraint (continuation is a constraint).
3. There is no notion of order invariant. Of course, the idea is
suggested by the definition of continuation.

BUT: the notion of order invariant set is quite fundamental, and
robust, and the order invariant and order theoretic sets are very
familiar to lots of mathematicians. See [1] for considerable
discussion and definitions involving sets fixed under automorphisms of
(Q,<).

WE NOW take aim at that equation at the end: S_1...n|>n =
S_0...n-1|>n. Of course, this is the kind of equation that is highly
beloved by anybody working or working part time in set theory, and
highly beloved by many combinatoricians focused on Ramsey theory.

Nevertheless, I have  been talking to algebraists and analysts, and
their immediate reaction is to find this equation "strange".

I have another way of saying this equation. This is in terms of
partial self embeddings. All through math one has the notion of self
embedding

EVERY ORDER INVARIANT SUBSET OF Q^2k HAS A MAXIMAL NONNEGATIVE ROOT,
SELF EMBEDDED BY +1 ON {0,...,n-1}, IDENTITY ON Q|>n.

EVERY FINITE SUBSET OF Q^k|>n HAS A MAXIMAL NONNEGATIVE CONTINUATION,
SELF EMBEDDED BY +1 ON {0,...,n-1}, IDENTITY ON Q|>n.

The notion of partial embedding is defined in [1]. h:Q into Q is
one-one partial, and for all p_1,...,p_k in dom(h),
(h(p_1),...,h(p_k)) in S iff (p_1,...,p_k) in S.

In [1], there is a complete determination of the partial h:Q into Q
which move finitely many points, that can be used in the first
statement above. These are the h that are strictly increasing.

For the second statement above, we also have a complete determination
of the partial h:q into Q which move finitely many points, that can be
used here. These are the h that are strictly increasing, and where all
points moved are <= n.

Note how I have now conveniently eliminated any use of letters for
either the given finite set or the continuation, in these statements.

I now am going back to my algebraists and analysts to see if they
prefer the embedding form above to the previous one with the equation.

The more general notion of r-continuation is mentioned in [1], [2].
The notion of continuation is just 2-continuation. Of course,
1-continuation leads to trivialities - just the use of the ordinary
Ramsey theorem.

EVERY FINITE SUBSET OF Q^k|>n HAS A MAXIMAL NONNEGATIVE
r-CONTINUATION, SELF EMBEDDED BY +1 ON {0,...,n-1}, IDENTITY ON Q|>n.

We now come to Finite Continuation Theory. We will close this posting
with a not very imaginative way of going about this, where we again
continue a finite set A of vectors to a set S of vectors. We are now
thinking about continuing a partial function to a partial function,
and if this comes out really satisfactory, then it will start to
realize my hopes for Continuation Theory.

Here we can move everything from Q to N (the nonnegative integers),
which is an advantage. There is a cost, which is that the preferred
elements cannot be taken to be 0,...,n. But they can be taken form an
arithmetic progression, which is fine. We will still be using self
embeddings by an explicitly given partial function.

DEFINITION 1. Let A,S be nonempty finite subsets of N^k and B be a
finite subset of Q|>=0. S is a B-maximal nonnegative continuation of A
if and only if
i. S is a nonnegative continuation of A, where  sup(fld(S)) = sup(fld(A)).
ii. S is not a proper subset of any nonnegative continuation of A
included in S union B^k.

LET t = (8knm)!!. EVERY E CONTAINEDIN {2kt,...,2kt+m}^k HAS A
{0,t,2t,...,nt}-MAXIMAL NONNEGATIVE CONTINUATION S WITH A
fld(S)-MAXIMAL NONNEGATIVE CONTINUATION SELF EMBEDDED BY +t on
{0,t,2t,...,(k-1)t}, IDENITITY ON {kt+1,...,2kt+m}^k.

Note that the above is explicitly Pi01. It is provably equivalent to
Con(SRP) over EFA.

************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 583rd in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory  1/27/15  10:39 AM
577: New Pi01 Incompleteness  3/7/15  2:54PM
578: Provably Falsifiable Propositions  3/7/15  2:54PM
579: Impossible Counting  5/26/15  8:58PM
580: Goedel's Second Revisited  5/29/15  5:52 AM
581: Impossible Counting/more  6/2/15  5:55AM
582: Link+Continuation Theory  1  6/21/15  5:38PM

Harvey Friedman